Azimuthal and Particle in a box equations

[maxiee]

Homework Statement

Mathematically, the Azimuthal equation is the same differential equation as the one for a particle in a box. But $$\Phi(\phi)$$ for $$m_l = 0$$, is a constant and is allowed, whereas such a constant wave function is not allowed for a particle in a box. What physics accounts for the difference?

Homework Equations

The Azimuthal Equation:
$$\frac{\partial ^{2} \Phi(\phi)}{\partial \phi^{2}} = -m_l ^{2} \Phi(\phi)$$

The particle in a box equation:
$$\frac{\partial ^{2} \psi(x)}{\partial \psi^{2}} = -k ^{2} \psi(x)$$

The Attempt at a Solution

The boundary conditions seem to play a role in the different allowed wave functions. However, I am having trouble relating the boundary conditions to the allowed quantum numbers.

Thanks in advance

Edit: The Azimuthal Equation corresponds to the Azimuthal motion of a particle. It comes about from the 3D Schrödinger Eq.
The Equation for a particle in a box is the result of the 1D Schrödinger Eq.

 

[kloptok]

There may come restrictions on $$m_l$$ or $$k$$ from the differential equations for the other coordinates. Now, I don't really remember the details but I think the radial equation imposes constraints on $$m_l$$. The notation implies that $$m_l=m_l(l)$$. That is, $$m_l$$ is a function of $$l$$, which appears in the equation for $$\theta$$ I think, solved by the Legendre polynomials. I might be remembering this wrong however, mixing things up. But in general, I think this would be the difference - that you have different constraints in the two cases.
[STRIKE]
It would probably be good if you could explain a bit more in detail where the equations come from, I can only assume you are solving some PDE similar to the wave eq. or Schrödinger eq.[/STRIKE]

EDIT: Sorry, I can see that it is most probably the Schrödinger eq. you are solving.